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Biconvex Biclustering

Sam Rosen, Eric C. Chi, Jason Xu

Abstract

This article proposes a biconvex modification to convex biclustering in order to improve its performance in high-dimensional settings. In contrast to heuristics that discard a subset of noisy features a priori, our method jointly learns and accordingly weighs informative features while discovering biclusters. Moreover, the method is adaptive to the data, and is accompanied by an efficient algorithm based on proximal alternating minimization, complete with detailed guidance on hyperparameter tuning and efficient solutions to optimization subproblems. These contributions are theoretically grounded; we establish finite-sample bounds on the objective function under sub-Gaussian errors, and generalize these guarantees to cases where input affinities need not be uniform. Extensive simulation results reveal our method consistently recovers underlying biclusters while weighing and selecting features appropriately, outperforming peer methods. An application to a gene microarray dataset of lymphoma samples recovers biclusters matching an underlying classification, while giving additional interpretation to the mRNA samples via the column groupings and fitted weights.

Biconvex Biclustering

Abstract

This article proposes a biconvex modification to convex biclustering in order to improve its performance in high-dimensional settings. In contrast to heuristics that discard a subset of noisy features a priori, our method jointly learns and accordingly weighs informative features while discovering biclusters. Moreover, the method is adaptive to the data, and is accompanied by an efficient algorithm based on proximal alternating minimization, complete with detailed guidance on hyperparameter tuning and efficient solutions to optimization subproblems. These contributions are theoretically grounded; we establish finite-sample bounds on the objective function under sub-Gaussian errors, and generalize these guarantees to cases where input affinities need not be uniform. Extensive simulation results reveal our method consistently recovers underlying biclusters while weighing and selecting features appropriately, outperforming peer methods. An application to a gene microarray dataset of lymphoma samples recovers biclusters matching an underlying classification, while giving additional interpretation to the mRNA samples via the column groupings and fitted weights.

Paper Structure

This paper contains 17 sections, 7 theorems, 59 equations, 5 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Biconvex Formulation
  3. Methods
  4. Convergence
  5. Adaptive BCBC
  6. Hyperparameter Selection
  7. Finite-Sample Bounds for Biconvex Biclustering
  8. Empirical Performance
  9. Varying Uninformative Features
  10. Varying Noise Magnitude and Dimension
  11. Genomics Case Study
  12. Discussion
  13. BCBC for Missing Data
  14. Kurdyka-Ł ojasiewicz Property of the Objective
  15. Results for Connected Affinity Graphs
  16. ...and 2 more sections

Key Result

Proposition 1

The minimizers of subproblems eq:palm1, eq:palm2 applied to the objective function eqn:objective_short are given by solving a biconvex clustering problem and projecting onto a weighted simplex, respectively. $\blacktriangleleft$$\blacktriangleleft$

Figures (5)

  • Figure 1: Mean of true bicluster ARI for each algorithm under the scenario described in Section \ref{['section:simulation1']}. The standard error is used to calculate 95% confidence bands for each point.
  • Figure 2: Distribution of AUC scores when fit under the scenario described in Section \ref{['section:simulation1']}.
  • Figure 3: Results for simulations described in Section \ref{['section:simulation2']}: Mean true bicluster ARI. Metrics in bold are the best performance across all algorithms for the given noise variance and matrix size. The standard error is displayed below each mean in parenthesis.
  • Figure 4: Distribution of AUC scores for models under the scenario described in Section \ref{['section:simulation2']}.
  • Figure 5: Biclustering results of Lymphoma data with top eight column clusters by mean weight outlined. The cDNA clone true labels correspond to DLBCL, FL, and CLL for colors cyan, green, and yellow, respectively. Blocks are labeled to be matched with Table \ref{['tab:biclusters']}.

Theorems & Definitions (15)

  • Proposition 1
  • proof
  • Proposition 2
  • Theorem 1
  • Definition 1: Desingularizing Function
  • Definition 2: KL property and KL function
  • Definition 3: Semialgebraic sets and functions
  • Proposition 3
  • proof
  • Lemma 1: Affinity Graph Topology
  • ...and 5 more